Problem: Which of the following functions have inverses?  Note that the domain of each function is also given.

A. $a(x) = \sqrt{2 - x},$ $x \in (-\infty,2].$

B. $b(x) = x^3 - x,$ $x \in \mathbb{R}.$

C. $c(x) = x + \frac{1}{x},$ $x \in (0,\infty).$

D. $d(x) = 2x^2 + 4x + 7,$ $x \in [0,\infty).$

E. $e(x) = |x - 2| + |x + 3|,$ $x \in \mathbb{R}.$

F. $f(x) = 3^x + 7^x,$ $x \in \mathbb{R}.$

G. $g(x) = x - \frac{1}{x},$ $x \in (0,\infty).$

H. $h(x) = \frac{x}{2},$ $x \in [-2,7).$

Enter the letters of the functions that have inverses, separated by commas.  For example, if you think functions $b(x)$ and $e(x)$ have inverses, enter "B, E" without the quotation marks.
A. The function $a(x) = \sqrt{2 - x}$ is decreasing, so it has an inverse.

B. Note that $b(0) = b(1) = 0,$ so the function $b(x)$ does not have an inverse.

C. Note that $c \left( \frac{1}{2} \right) = c(2) = \frac{5}{2},$ so the function $c(x)$ does not have an inverse.

D. The function $d(x) = 2x^2 + 4x + 7 = 2(x + 1)^2 + 5$ is increasing on $[0,\infty),$ so it has an inverse.

E. Note that $e(2) = e(-3) = 5,$ so the function $e(x)$ does not have an inverse.

F. Both $3^x$ and $7^x$ are increasing, so $f(x) = 3^x + 7^x$ is also increasing.  Hence, it has an inverse.

G. Suppose $g(a) = g(b)$ for some $a,$ $b > 0.$  Then
\[a - \frac{1}{a} = b - \frac{1}{b}.\]Multiplying both sides by $ab,$ we get
\[a^2 b - b = ab^2 - a.\]Then $a^2 b - ab^2 + a - b = 0,$ which factors as $(a - b)(ab + 1) = 0.$  Since $a$ and $b$ are positive, $ab + 1$ cannot be 0, so $a = b.$

We have shown that $g(a) = g(b)$ forces $a = b,$ so the function $g(x)$ has an inverse.

H. The function $h(x) = \frac{x}{2}$ has an inverse, namely $h^{-1}(x) = 2x.$

Thus, the letters of the functions that have inverses are $\boxed{\text{A, D, F, G, H}}.$